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Fix $0<a\le 1$ and $-1\le b\le 1$. A single deterministic quantum channel sending every state of vector $a\mathbf{n}$ to the state of vector $b\mathbf{n}$ exists exactly under the displayed condition. Add the requirements for $\mathbf{n}$ and $-\mathbf{n}$ to show that the channel fixes $I$. Subtract and take the three coordinate directions to get its action on each Pauli matrix, namely multiplication by $b/a$. Linearity thus uniquely forces the isotropic qubit map ${\mathcal{E}}_{b/a}$. Its complete-positivity range proves necessity and sufficiency. A basis-dependent unitary may reverse one known vector without implementing a universal inversion, so knowing the basis changes the conversion problem.

Conjugation by one Pauli matrix fixes that same matrix and reverses the other two. For example, $XXX=X$, $YXY=-X$, and $ZXZ=-X$. Applying this to each component of the Bloch vector givesHence the quantum depolarizing channel isThe Kraus operators $\sqrt{1-p}I$ and $\sqrt{p/3}{\sigma }_j$ establish complete positivity for $0\le p\le 1$. At $p=3/4$ every state becomes maximally mixed. For $p>3/4$ the factor is negative: it reverses the vector as well as reducing its length. Its most negative value is $-1/3$, consistent with the complete-positivity range of the isotropic qubit map.

The universal NOT gate is the isotropic qubit map with factor $-1$. It sendsFor $|\mathbf{s}|\le 1$, the output has the same two nonnegative eigenvalues as the input, exchanged, and still has trace one. Thus it maps every single-qubit density matrix to a density matrix. Its linear extension to all operators is $\mathcal{N}(A)=\mathrm{tr}(A)I-A$, not $I-A$ for arbitrary unnormalized $A$. Positivity on one qubit does not establish complete positivity when an ancilla is present.

By linearity, the isotropic qubit map acts asIt contracts or expands every direction by the same factor and reverses the vector when $r<0$. More generally ${\mathcal{E}}_r(A)=rA+(1-r)\mathrm{tr}(A)I/2$, so the map preserves trace and Hermiticity for every real $r$.

The ideal universal NOT map sends every qubit Bloch vector $\mathbf{s}$ to $-\mathbf{s}$, taking a pure state to its orthogonal pure state. It is a trace-preserving positive map but not a quantum channel: applied to one half of a Bell state, it produces an operator with a negative eigenvalue. A physical approximation can reverse all directions with factor $-1/3$ using the equal mixture of conjugations by the three Pauli matrices. The isotropic qubit map complete-positivity bound proves that a larger uniform reversal is impossible among such isotropic maps.